Kappos 2002 dynamic loading and design of structures

Page i Dynamic Loading and Design of Structures Page ii This page intentionally left blank Page iii Dynamic Loading and Design of Structures Edited by A.J.Kappos London and New York Page iv First published 2002 by Spon Press 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Spon Press 29 West 35th Street, New York, NY 10001 Spon Press is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2004 © 2002 Spon Press All rights reserved No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Dynamic loading and design of structures/edited by A.J.Kappos p cm Includes bibliographical references ISBN 0-419-22930-2 (alk paper) Structural dynamics Structural design I.Kappos, Andreas J TA654.D94 2001 624.1'7–dc21 2001020724 ISBN 0-203-30195-1 Master e-book ISBN ISBN 0-203-35198-3 (OEB Format) ISBN 0-419-22930-2 (Print Edition) Page v Contents List of contributors ix Preface xi Probabilistic basis and code format for loading MARIOS K.CHRYSSANTHOPOULOS Introduction Principles of reliability based design 1 Framework for reliability analysis 10 Time-dependent reliability 13 Actions and action effects on structures 19 Concluding remarks 28 References 29 Analysis for dynamic loading GEORGE D.MANOLIS Introduction 31 31 The single degree-of-freedom oscillator 31 Multiple degree-of-freedom systems 46 Continuous dynamic systems 56 Base excitation and response spectra 58 Software for dynamic analysis 64 References 64 Wind loading T.A.WYATT Wind gust loading 67 67 Aerodynamic instability 81 Aeroelastic excitation 98 References 105 Page vi Earthquake loading ANDREAS J.KAPPOS Introduction 109 109 Earthquakes and seismic hazard 109 Design seismic actions and determination of action effects 125 Conceptual design for earthquakes 160 References 171 Wave loading TORGEIR MOAN Introduction 175 175 Wave and current conditions 177 Hydrodynamic loading 186 Calculation of wave load effects 198 Dynamic analysis for design 210 References 226 Loading from explosions and impact ALAN J.WATSON Introduction 231 231 Blast phenomena 233 Impact phenomena 246 Design actions 253 Designed response 262 Damage mitigation 272 Design codes 276 References 282 Human-induced vibrations J.W.SMITH Introduction 285 285 The nature of human-induced dynamic loading 286 Methods for determining the magnitude of human-induced loading 291 Design of structures to minimise human-induced vibration 303 References 304 Traffic and moving loads on bridges DAVID COOPER Introduction Design actions 307 307 308 Determination of structural response 311 References 322 Page vii Machine-induced vibrations J.W.SMITH Introduction 323 Dynamic loading by machinery 324 Design of structures to minimise machine-induced vibration 331 References 341 10 Random vibration analysis GEORGE D.MANOLIS Introduction Index 323 343 343 Random processes 344 System response to random input 350 Structures with uncertain properties 363 References 367 369 Page viii This page intentionally left blank (10.91) This result was encountered in the earlier part of this section in conjunction with the Fokker— Planck equation If the restoring force g(x) is split into a linear and a nonlinear part according to eqn (10.78), then: (10.92) and a perturbation technique needs to be employed Page 362 10.3.5 Example: non-stationary case As an example, consider the simple case of a SDOF system with a finite operating time t0=0 subjected to a stationary random process Although the input is stationary, the output is not, by virtue of the fact that the system has a finite operating time Consider therefore eqn (10.27) under zero initial conditions and where input f(t) is a member function of a zero mean stochastic process which is stationary, ergodic and described by a PSDF equal to Sff(ω) First we have that the output process x(t) also has a zero mean, as can be seen by recourse to eqn (10.32) Next, the variance of x(t) is (see eqn (10.33)): (10.93) Sincef(t) is stationary and this autocorrelation function is related to the PSDF via the Wiener—Khinchine relation (Caughey, 1963, 1971) as: (10.94) where it is assumed that Sff (ω) dω< Using eqn (10.94) in eqn (10.93) gives: (10.95) Since the integrals involved in the above equation are convergent, the order of integration may be reversed Using the definition of h(t) in eqn (10.30) and carrying out the integrations gives: (10.96) where |H(ω)|−2 can be found by recourse to eqn (10.68) as: (10.97) As t→ ∞in eqn (10.96), as expected Furthermore, as t→ ∞, , a result in agreement with harmonic (i.e steady state) analysis of the SDOF system that was also recovered in conjunction with eqn (10.87) Finally, a common approximation for a lightly damped SDOF system is to set Sff(ω)=2S0/π, as shown in Figure 10.8 In that case, Page 363 Figure 10.8 Random SDOF system response , a result for stationary conditions that can be found in many references (Hinch, 1991) 10.4 STRUCTURES WITH UNCERTAIN PROPERTIES 10.4.1 Static analysis So far, we have examined the case where a structure is deterministic and its excitation is random We will now look at a FEM formulation for stochastic cases where randomness can be expressed in the general form , with z0 being the deterministic value of a material property (such as the elastic modulus) or a structural component (such as the moment of inertia of a member) and γbeing a random, zero mean small fluctuation about z0 Following (Vanmarke et at., 1986), we will utilize the FEM stiffness approach which gives the following system of algebraic equations for the static case: (10.98) As before, [K] is the N×N stiffness matrix and {x} and {f} are N×1 column vectors containing the nodal displacements and forces, respectively The important distinction to be made here is that the uncertainty in the structure is reflected in the stiffness matrix and, upon solution, on the nodal displacements Also, since the case of random input was examined in the previous sections, {f} is assumed to be deterministic here The stiffness matrix can now be expanded about the uncertainty using Taylor series as: (10.99) Page 364 where n denotes the total number of random parameters γ i As before, superscript denotes a deterministic quantity, while the next two terms in the expansion respectively denote first and second rates of change which are evaluated by differentiating [K] with respect to the random parameters γ i Note that the use of commas indicates partial differentiation with respect to the subscript that follows The same type of expansion can also be used for the displacements, that is: (10.100) where the range of the summation indices is omitted for reasons of notational convenience Substitution of the above two expansions in eqn (10.98) and a subsequent perturbation-type ordering of the terms gives the following system of equations: (10.101) The structure of the above system of equations is similar to that of eqn (10.80) which was obtained for non-linear systems in Section 10.3.4 using perturbations Thus, all unknown displacement terms can be obtained sequentially, starting from the deterministic solution {x0} and substituting the newly found terms in the right-hand side of the next equation As a result, the deterministic stiffness matrix needs to be inverted only once, resulting in an efficient solution scheme Also, the non-zero terms in [K] ;i and [K],ij are relatively few so that the right-hand sides can be quickly formed The same approach can be used for problems involving lack of fit in structural members by introducing the concept of initial strains, as well as for structures on an elastic foundation with uncertain foundation modulus by introducing the foundation reaction matrix Finally, uncertainty in the boundary conditions can be accounted for by inserting virtual springs at the boundaries and taking the spring constants as uncertain Following the displacement solution, the unknown stress tensor on any point within a finite element can be found after the stress terms {σ0}, {σ},i and {σ},ij have been evaluated in the usual way from their corresponding displacement terms {x0}, {x},i and {x}ij.Thus, the final expression for the stress tensor is: (10.102) Based on the above equation, the expectation and variance of the stresses are: (10.103) Page 365 and (10.104) respectively The second moments of the random variables γ i are related to the power spectrum Sγγ(k) via the Wiener—Khinchine relation (Lampard, 1954) as: (10.105) where r is the distance between nodal co-ordinates and k denotes the wave number Since local changes in a structural parameter cause non-linear changes in the structural response, a second order Taylor series expansion such as the one used here is necessary to cover such non-linearities Third order expansions are preferable, but computation becomes prohibitively expensive since sixth moments of the random variables γ i are necessary for compatibility in the computation of stress variances 10.4.2 Dynamic analysis As a first step, we consider the eigenvalue problem: (10.106) where [M] is the mass matrix, λare the eigenvalues and {φ} are the eigenvectors As before, uncertainty in the stiffness and mass matrices filters, upon solution, to the eigenproperties of the structure We begin (Liu et al., 1986) by expanding both eigenvalues and eigenvectors in a Taylor series about the randomness γas: (10.107) and (10.108) respectively Substitution of the above two expressions in eqn (10.106) along with eqn (10.99) and a similar expansion for the mass gives, after the usual perturbation-type ordering, the following system of equations: Page 366 (10.109) By taking advantage of symmetry in [H0], λ,i can be computed from the second of eqns (10.109) as: (10.110) Determination of {φ}i from the second of eqns (10.109) is not, however, feasible because of the singularity of [H0] To overcome this drawback, a reduction in the rank of [H0] is necessary The same situation holds for the evaluation {φ},ij of since only the right-hand side of eqns (10.109) changes As with the static case, each new eigenvalue solution depends on the previously obtained eigenproperties As far as time history analyses are concerned, the most rational approach is to go to a modal co-ordinate environment and assume that properties such as the modal damping ratios ζ are uncertain Although this ignores the fact that uncertainty is first manifested at the physical co-ordinate level in terms of uncertain stiffness and mass, the convenience of decoupled modal equations is too tempting to ignore The analysis at the modal co-ordinate level is essentially the same as the perturbation approach used in Section 10.3.4 for a non-linear SDOF system under random input In particular, the modal damping ratio is written as: (10.111) where i is a modal DOF, while the modal co-ordinate yi is expanded as (10.112) where superscripts (1), (2) on y respectively denote first and second order perturbation terms which are random processes Substitution of the above two expansions in the ith uncoupled equation of motion (see eqn (10.39)) gives the following system of equations: (10.113) Page 367 Numerical integration of the above system can proceed without difficulties Following solution for all expansion terms of yi(t), one may return to physical co-ordinates (see eqn (10.37)) and apply the statistical averaging using the expectation to find the response statistics 10.5 REFERENCES Augusti, G., Barrata, A and Casciati, F (1984) Probabilistic Methods in Structural Engineering, Chapman and Hall, London Caughey, T.K (1963) ‘Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white noise excitation’, Journal of the Acoustical Society of America 35(11) : 1683–92 Caughey, T.K (1971) ‘Nonlinear theory of random vibrations’, Advances in Applied Mechanics 11:209–53 Coddington, E.A and Levinson, N (1955) Theory of Ordinary Differential Equations, McGraw-Hill, New York Crandall, S.H (1963) ‘Zero crossings, peaks and other statistical measures of random response’, Journal of the Acoustical Society of America 35(11): 1693–9 Crandall, S.H and Mark, W.D (1963) Random Vibration in Mechanical Systems, Academic Press, New York Ghanem, R and Spanos, P.D (1991) Stochastic Finite Elements: A Spectral Approach, SpringerVerlag, New York Hinch, E.J (1991) Perturbation Methods, Cambridge University Press, Cambridge Hurty, W.C and Rubinstein, M.F (1964) Dynamics of Structures, Prentice-Hall, Englewood Cliffs, NJ Klieber, M and Hien, T.D (1992) The Stochastic Finite Element Method, John Wiley, New York Lampard, D.G (1954) ‘Generalization of the Wiener—Khintchine theorem to nonstationary processes’, Journal of Applied Physics 25:802–3 Liu, W.K., Belytscko, T and Mani, A (1986) ‘Random field finite elements’, International Journal for Numerical Methods in Engineering 23:1831–45 Nigam, N.C (1983) Introdution to Random Vibrations, MIT Press, Cambridge, MA Vanmarke, E., Shinozuka, M., Nakagiri, S., Schueller, G.I and Grigoriou, M (1986) ‘Random fields and stochastic finite elements’, Structural Safety 3:143–66 Zayed, A.I (1996) Handbook of Function and Generalised F unction Transforms, CRC Press, Boca Raton, FL Page 368 This page intentionally left blank [...]... the outcome of the valuable research carried out in the various fields included under ‘Dynamics’, make this type of analysis a part of everyday life in the design office There are also a number of good reasons why dynamical behaviour of buildings, bridges and other structures is now more of a concern for the designer than it used to be 20 or 30 years ago One reason is that the aforementioned structures. .. lack of a book dealing with all types of dynamic loading falling within the scope of current codes of practice, makes the problem even more acute The main purpose of this book is to present in a single volume material on dynamic loading and design of structures that is currently spread among several publications (books, journals, conference proceedings) The book provides the background for each type of. .. ensuring an acceptable range of safety level and an acceptable economy of construction Hence, it is often useful to make the distinction between primary basic variables and other basic variables The former group includes those variables whose values are of primary importance for design and assessment of structures The above concepts of characteristic and design values, and associated partial factors,... the various clauses of the code lead to the desirable level of safety of structures designed to the code It relates to the number of design checks required, the rules for load combinations, the number of partial factors and their position in design equations, as well as whether they are single or multiple valued, and the definition of characteristic or representative values for all design variables Page... overseas, and involved in the design of civil engineering structures for various types of dynamic loads Depending on the type of loading addressed, an attempt was made to present code provisions both from the European perspective (Eurocodes, British Standards) and the North American one (UBC, NBC), so the book should be of interest to most people involved in design for dynamic loading worldwide The book... also aims at research students (MSc and PhD programmes) working on various aspects of dynamic loading and analysis With regard to MSc courses, it has to be clarified that Loading is typically a part of several, quite different, courses, rather than a course on its own (although courses like Loading and Safety’ and ‘Earthquake Loading , do currently exist in the UK and abroad) This explains to a certain... include people from leading design firms and/ or with long experience in the design of structures against dynamic loads Putting together and working with the international team of authors that was indispensable for writing a book of such a wide scope, was a major challenge and experience for the editor, who would like to thank all of them for their most valuable contributions Some of the contributors, as... (Chapter 8), and machinery (Chapter 9) In each chapter the origin of the corresponding dynamic loading is first explained, followed by a description of its effect on structures, and the way it is introduced in their design The latter is supplemented by reference to the most pertinent code provisions and an explanation of the conceptual framework of these codes All these chapters include long lists of references,... The dynamical behaviour of civil engineering structures has traditionally been tackled, for design purposes, in an ‘equivalent static’ way, essentially by introducing magnification factors for vertically applied loads and/ or by specifying equivalent horizontal loads Today the availability of software able to deal explicitly with dynamic analysis of realistic structures with many (dynamic) degrees of. .. probabilistic and reliability methods have been developed to help engineers tackle the analysis, quantification, monitoring and assessment of structural risks, undertake sensitivity analysis of inherent uncertainties and make rational decisions about the performance of structures over their working life These tasks may be related to a specifie structure, a group of similar structures or a larger population of structures ...Page i Dynamic Loading and Design of Structures Page ii This page intentionally left blank Page iii Dynamic Loading and Design of Structures Edited by A.J .Kappos London and New York Page... Library of Congress Cataloging in Publication Data Dynamic loading and design of structures/ edited by A.J .Kappos p cm Includes bibliographical references ISBN 0-419-22930-2 (alk paper) Structural dynamics... nature of human-induced dynamic loading 286 Methods for determining the magnitude of human-induced loading 291 Design of structures to minimise human-induced vibration 303 References 304 Traffic and